Redundancy in a Tuning System

Redundancy is one look which makes a tuning system a tuning system.

Gregory Bateson has this to say about redundancy in his article "Style, Grace, and Information in Primitive Art":

Any aggregate of events or objects (e.g., a sequence of phonemes, a painting, or a frog, or a culture) shall be said to contain "redundancy" or "pattern" if the aggregate can be divided in any way by a "slash mark," such that an observer perceiving only what is on one side of the slash mark can guess, with better than random success, what is on the other side of the slash mark. We may say that what is on one side of the slash contains information or has meaning about what is on the other side. Or, in engineer's language, the aggregate contains "redundancy."

I (- Andrew_Heathwaite Nov 12, 2009) assert:

• A system (eg. "a work of art", "a composition", "a tuning system") implies a network of redundancies.
• We can talk about these redundancies: "less redundancy"; "more redundancy"; "no redundancy", etc.
  • "Less" or "more" or "no" redundancy is determined by each listener (largely unconsciously) according to her structure at that moment.
• [Perceived] more redundancy in one aspect may ground a piece so that [perceived] less redundancy in another aspect can be seen as meaningful.
  • "One thing stays the same so another thing can change."
  • "One thing changes so another thing can stay the same."
• "No redundancy" can be seen as redundant when the listener can predict continued lack of redundancy.
  • "I know what it's going to do next - something random!"
  • A listener may hear or not hear (make or not make) an intended redundancy/pattern.
• Answers to questions of which redundancies (or which networks of redundancies) help us decide things like "style", "meaning", "how should I listen to this?" and "is this music, anyway?"

If I look a tuning as a system, I might consider the tuning's "redundancy."
How does this tuning system exhibit pattern?
I care about how a tuning system offers redundancy so that I may choose a different network of redundancies than th one(s) I'm familiar with (a fish attempting to "see" the water she lives in).

Here is a far from complete list of aspects of tunings systems that might indicate redundancy of some kind. Please add to it!

• the use of definite pitches
  • instead of, or in addition to noise
• a sustained pitch
  • redundancy in time
• a scale (set of definite pitches)
  • redundancy in time: "I've heard these pitches before"
  • less redundant: a scale which changes (for instance, ascending different from descending)
• harmony influenced by a theory of consonance
  • more than one pitch sounding together & their distinctness blurring
  • if pitch 'y' appears in the overtone series of pitch 'x', when pitch 'x' is sounding (on a harmonic instrument), pitch 'y' will also sound (although might not be identifiable as a distinct pitch) - so building a scale which contains both 'x' & 'y' is redundant
• octave equivalence
  • B-flat considered equivalent to B-flat an octave higher...
  • an octave-repeating scale
    • the "same" pitches are available in each octave
• equal divisions of some interval
  • less redundant: maximally even a.k.a. "quasi-equal" divisions
• tuning lattices
  • "two-dimensional", "three-dimensional", etc. where each axis represents iterations of a single interval
• harmonic limit
• building scales from/with tetrachords
  • more redundant: the same tetrachord repeated in the gamut
  • less redundant: two (or more) different tetrachords in the gamut
• moment of symmetry technique (making a chain of one 'generator' interval, folded within a set 'period' interval)
  • produces scales with only two step sizes & only two intervals of each class
  • less redundant extensions: MOS Cradle or Second Order MOS
• matching tuning to timbre
  • eg. rational intonation - selecting pitches which are copied in the overtone series of other intervals
  • eg. Indonesian gamelan tuning - the scale is made to match the inharmonic pitches of the metals used in building the instruments
  • octaves in the high range of a piano are routinely stretched to fit with the "distorted" sound of the high-tension strings
• matching timbre to tuning
  • the reverse of the above: making a timbre (usually with a computer) to "match" a stipulated tuning